Lemma 2.1.3
Let
Proof of 1:
It follows from the 2.1.13 Theorem (Continuous functional calculus) That if
This implies that
For each
Since
Since the map is continuous for t at every x, in particular it should be continuous at just h
Therefore
Proof of 2:
If
Then
Set
The main idea here is that if the element has as spectrum only part of the unit circle, we can define a branch on the argument function acting on the spectrum of the element, from there we can access our element as an argument of some self adjoint element on the unit circle, then we apply part 1 to this exponential.
Proof of 3:
If
This in turn implies that
This proof exploits the relationship between the spectral radius of a normal element and its norm, along with the convenient fact that
$v^{*}u\sim{h}1$ and $v\sim{h}v\implies v(v^{*}u)\sim_{h}v$ .